I'm having trouble wrapping my head around what should be a trivial detail, but it is important, so hopefully someone else putting it in explicit words might help me understand it. What I am having trouble grasping is why do row operations preserve linear dependence relations for the columns of a matrix but not the rows? The context this comes up is in regards to the row & column space of a matrix. Given a matrix A, to find a basis for the column space we would just take the linearly independent columns of A. However, usually it's difficult to tell what columns are independent, so we find rref(A) and the pivot positions in rref(A) correspond directly to the pivot positions in A, this is true because row operations preserve linear dependence relations for the columns. For the row space we would take the linearly independent rows of rref(A) this is because the row space of A is equivalent to rref(A); however, the dependence relations for the rows are not the same. So if I can understand why the dependence relations are the same for columns but different for rows, it would really help me connect everything together.